Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ration 2:1 (i) internally (ii) externally

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Solution

(i)

The given position vectors are OP → = i ^ +2 j ^ − k ^ and OR → =− i ∧ + j ∧ + k ∧ .

The position vector of R dividing P and Q internally in the ratio of 2:1 is,

OR → = 2× OQ → +1× OP → 2+1 = 2×( − i ^ + j ^ + k ^ )+1×( i ^ +2 j ^ − k ^ ) 3 = − i ^ +4 j ^ + k ^ 3

Thus, the position vector of R dividing P and Q internally in the ratio of 2:1 is −1 3 i ^ + 4 3 j ^ + 1 3 k ^ .

(ii)

The given position vectors are OP → = i ^ +2 j ^ − k ^ and OR → =− i ∧ + j ∧ + k ∧ .

The position vector of R dividing P and Q externally in the ratio of 2:1is,

OR → = 2× OQ → −1× OP → 2−1 = 2×( − i ^ + j ^ + k ^ )−1×( i ^ +2 j ^ − k ^ ) 1 =−3 i ^ +3 k ^

Thus, the position vector of R dividing P and Q externally in the ratio of 2:1 is −3 i ^ +3 k ^ .

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Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ration 2:1

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